I was given an excersice to solve wich asks you to prove:
$$\int_0^1 f(r) r dr = 0 $$
knowing that:
$$\int_0^1 f(t) dt = 0 $$
After doing integration by parts I ended up with:
$$\int f(r) r dr = (\int f(r)dr)r- \int\int f(r)drdr $$
My question is, how should I proceed in order to evaluate the integral with my integration limits between 0 and 1. I tried with Barrow, but I don't know what to do with the double integral and its limis